Courtecuisse & Duhem 1995 Mushrooms & toadstools of Britain and Europe treat Morchella costata as "confused with M. elata Fr:Fr (=M. conica) which has less parallel ribs … (Habitat is different as well and different from our case as well!) Who knows what it is, but Morchella cf. confusa is the best designation of this collection at this moment.
Mind you, I do not know how is Morchella costata handled in the modern taxonomical mess of the Morchella genus.

P.S. Since the very beginning of using MO, I am trying to convince MO developers that there is a difference between “Morchella” and “Morchella sp.” and that stripping the “sp.” from the observation name is a mistake (not speaking about the absurdity of using the authors of the genus instead!):
1) “Morchella” (without sp.) means that I have not looked at the collection, but guessed that it could be “Morchella sp.” when I would have a better look at it. 2) “Morchella sp.” on the other hand, would indicate that I am sure that it would be Morchella, but I did not end up with any “good” name when I tried to identify it.

Richard et al.¹ do not say why Morchella costata (Vent.) Pers. is illegitimate (and do not cite any ICN section). (And maybe I’ve been too hasty in deprecating it in MO.) Perhaps, with your knowledge of the ICN, you could shed some light on this.

MycoBank lists two names for Morchella costata: M. costata (Vent.) Pers. (1801), and M. costata J.C. Schmidt & Kunze (1817), and says they are both Legitimate!²

Index Fungorum also has two names, but:
1. IF lists the first as Morchella costata Pers. with the comment “Replaced synonym: Phallus costatus Vent. 1797”. So at first glance, the authority should be “(Vent.) Pers.”. But IF also says that the basionym, Phallus costatus Vent., Ann. Bot. (Usteri) 21: 510 (1797) is illegit. under ICN Art. 53.1. What’s the significance of that? What’s the proper authority for Morchella costata: “(Vent.) Pers.” or just “Pers.”?

2. In contrast to MB, IF says that M. costata J.C. Schmidt & Kunze (1817) is illegitimate (under ICN Art. 53.1). That makes sense given the earlier publication of M. costata (Vent.) Pers. (1801)/M. costata Pers. But see below.

Is there a technical defect in Persoon’s protologue for M. costata? Was he required to explicitly name “Phallus costatus Vent.”? (Persoon cites Ventenat, but doesn’t mention the name “Phallus costatus”. He says “Ventenat l.c.p. 510. sub Phallo”. Is that good enough?)

¹ If you want to read more, the Richard article is freely available at https://www.tandfonline.com/doi/full/10.3852/14-166.

² To see how MB lists both homonyms as Legitimate, do a MB Basic Name Search for the Taxon Name “Morchella costata”. The results include:

³ IF adds “Editorial comment: Phallus costatus Batsch 1783”. But that’s a later date, and furthermore is a synonym of Helvella crispa. Doesn’t seem relevant to the questions above.

I have to confess that I do not know what makes the name illegitimate. In the Code you have a funny definition that legitimate names are all those names that are not illegitimate (Article 6.5). Never mind, I don’t see any reason why we could not write “Morchella cf. costata”. I wonder if Richard et al. stated why Morchella costata is an illegitimate name. Do they cite the Code article by which that name become illegitimate?

Per Richard, et al.:

The name M. costata (Vent.) Pers., which is often used in European literature, is an illegitimate synonym of M. elata (see below, Doubtful names).