I am looking for an extension of a result of Riche-Soergel about a functor which maps Soergel bimodules to Soergel modules.
 
Fix a given Coxeter system $(W,S)$, together with a (reflection faithful) representation $\mathfrak{h}$ of $W$. As usual, let $R=\mathrm{Sym}(\mathfrak{h}^\ast)$ and consider the asssociated category of Soergel bimodules, denoted by $\mathbb{S}\mathrm{Bim}$, as well as the category of Soergel modules, denoted by $\overline{\mathbb{S}\mathrm{Bim}}$ . Soergel modules are defined as graded right $R$-modules which are (finite direct sums of shifts of) direct summands of Bott-Samelson modules 
$$ \overline{\mathrm{BS}}(s_1,\dots,s_k):=\mathbb{k}\otimes_R R\otimes_{R^{s_1}} \dots \otimes_{R^{s_k}} R$$
The functor $ \mathbb{k}\otimes_R (-):\mathbb{S}\mathrm{Bim}\to \overline{\mathbb{S}\mathrm{Bim}}$ has been studied by Soergel and it was shown that for finite Weyl groups, the map 
$$\mathbb{k}\otimes_R \mathrm{Hom}_{\mathbb{S}\mathrm{Bim}}^\bullet(B,B')\to\mathrm{Hom}_{\overline{\mathbb{S}\mathrm{Bim}}}^\bullet(\mathbb{k}\otimes_R B,\mathbb{k}\otimes_R B')$$
is an isomorphism. This result was extended by Riche-Soergel in the case of finite Coxeter groups. One can find the precise references for these proofs in Elias-Makisumi-Thiel-Williamson's book on Soergel bimodules at proposition 15.27.

> **Does the above isomorphism hold for more general Coxeter systems?** For example, is this result known to hold even in the case of infinite dihedral type? Is there a simple counter-example that I have yet to find? 

I've been digging through litterature and I can't seem to find anything. I should note that Riche-Soergel's proof makes use of the longest element of the Coxeter group, which means it is not easily extended to the general case.

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**Edit (August 2024).**

After a bit of reading, I found some counter-examples to the above isomorphism in L. Patimo's thesis which one can find [here][1]. The counter-examples are given in section 3.6. 

However, a reputable source told me that the result should still hold in the infinite dihedral case above. This leaves this seemingly much more subtle question :

> *When* does the above isomorphism hold? Is there a simple criterion distinguishing Coxeter systems for which it holds?

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Any help is appreciated.



  [1]: https://lpatimo.github.io/PhdThesis.pdf