Per POWO as of 2021, the taxon change would (still) be:

Bryomorphe aretioides --> Helichrysum sessilioides (POWO)

but Dolichothrix ericoides is accepted at POWO
as is Bryomorphe lycopodioides ...?

@botaneek @tonyrebelo any insight?

Careful:

POWO has:
Bryomorphe aretioides Druce = > Helichrysum sessilioides
Helichrysum aretioides Turcz. = > Bryomorphe lycopodioides
Bryomorphe lycopodioides (Sch.Bip.) Levyns is accepted

BODATSA has:
Bryomorphe aretioides (Turcz.) Druce Accepted (Indigenous; Endemic)

Bryomorphe lycopodioides (Sch.Bip. ex Walp.) Levyns Synonym to: Dolichothrix ericoides (Lam.) Hilliard & B.L.Burtt
Bryomorphe zeyheri Harv. Synonym to: Bryomorphe aretioides (Turcz.) Druce

"I would say POWO is probably confused?? " Yes... and not only with this plant-.-

Here is a paper dealing with it.
Basically iNaturalist and BODATSA have it correct, and POWO needs to update.
Which they probably will today (but it will have to wait for the update to go online - so dont bother posting deviations, we just need to remember to update the POWO links when done).

ASTERACEAE
BRYOMORPHE AND DOLICHOTHRIX (GNAPHALIEAE–RELHANIINAE): TAXONOMY AND NOMENCLATURE
M. KOEKEMOER*
Bothalia 41,2 (2011)

Turczaninow (1851) described Helichrysum aretioides from the specimen (Zeyher 2908) collected on the Cape Mountains. This Russian botanist and administrator, a civil servant in the Departments of Justice and Finances (Stafleu & Cowan 1986), collected actively in the areas where he served, and published mainly on the Russian and Chinese floras. He is relatively unknown in the history of southern African botany but was a contemporary of Cassini, De Candolle and Thunberg. He most probably received the Zeyher specimen from De Candolle, with whom he corresponded frequently (letters in G, Stafleu & Cowan 1986).

Harvey (1863) described a new genus, Bryomorphe, for this unusual plant but was not satisfied to have the sterile specimen of Zeyher 2908 in TCD as a type (duplicates in K, P, PRE, S, SAM and TCD) (acronyms according to Holmgren et al. 1990; electronic specimen seen, denoted by e!). He selected a second specimen, Roser 42, on which he based his description and illustration for B. zeyheri. In this description Harvey cited H. aretioides Turcz. as a synonym, which is based on the same type collection, Zeyher 2908. Unfortunately Harvey (1863) also cited Klenzea lycopodioides Sch.Bip. as a synonym. Harvey’s citation of these two earlier names makes his name illegitimate and superfluous.

Levyns (1942) therefore chose the oldest name and made the new combination, Bryomorphe lycopodioides (Sch.Bip.) Levyns, based on Klenzea lycopodioides Sch.Bip. It is very unlikely that she saw the specimen of Krauss before she made this combination because Bryomorphe and Dolichothrix are very distinct monotypic genera and can not be confused, even on very poor sterile material. The holotype of K. lycopodioides in the Paris Herbarium (Krauss s.n., Dist. George, Roodewal) clearly represents a plant we today know as Dolichothrix ericoides (Lam.) Hilliard & B.L.Burtt.

Druce (1917) made the correct combination, Bryomorphe aretioides (Turcz.) Druce, and he should be followed. Fairly recently, the name Helichrysum arctioides Turcz. has come into use and was taken up in Klopper et al. (2006) as a synonym of Dolichothrix ericoides. As far as I can establish, Turczaninow never described such a species and the name may just be the result of an orthographic error due to misreading ‘are’ as ‘arc’ in poor copies of the original text.

Is it not (is it ever?) so simple:

Rafaël Govaerts writes (19 January 2021):

I’m afraid the Bothalia paper makes no sense at all.

Bryomorphe was published with 1 name B. zeyheri so that MUST be the type (not B. aretioides published a century later)
as also stated in ING:

As also stated by ING, B. zeyheri is a superfluous name for Klenzea lycopodioides (as that is the oldest included name) so automatically (Art. 7.5.) the type specimen of B. zeyheri becomes the type specimen of Klenzea lycopodioides which also becomes the type specimen of the genus.
In the Bothalia paper they say that the type specimen of Klenzea lycopodioides is the same as Dolichothrix ericoides so by definition then also B. zeyheri and the genus Bryomorphe MUST be synonyms of Dolichothrix as they are all based on the same type.

I’m not sure if it can be argued that Harvey definitely indicated a different type by using the name “zeyheri” and therefore indicating the Zeyher specimen as the type as this is not an explicit exception in Art. 7.5. only clear indication of a replaced name 7.5.(a) or the use of an existing epithet 7.5.(b) are listed as definite indications of a different type.
If anyone has access to J. S. African Bot. 8: 283. Oct 1942, there may be more discussion there but as she made the combination B. lycopodioides she definitely did not consider the Zeyher specimen as the type.

I think the only solution would be to conserve the genus with a conserved type (or just describe a new genus).

Protologue:
https://www.biodiversitylibrary.org/page/2898328#page/35/mode/1up

@loarie - can you resolve this please.
We want a swap, not a name change ...

thanks