Identities in twisted Brauer monoids

The twisted Brauer monoid $\mathcal{W}_n$ is the monoid generated by $s_1,\dots,s_{n-1},h_1,\dots,h_{n-1},c$, subject to the following relations that hold for all $i,j=1,\dots,n-1$:
\begin{align*} &h_{i}h_{j}=h_{j}h_{i}, \ \ s_{i}s_{j}=s_{j}s_{i}, \ \ h_{i}s_{j}=s_{j}h_{i} &&\text{if } |i-j|\ge 2;\\ &h_{i}h_{j}h_{i}=h_{i}, \ \ s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j}, \ \ s_is_jh_i = h_js_is_j &&\text{if } |i-j|=1;\\ &h_{i}^2=ch_{i}=h_{i}c, \ \ s_{i}^2=1, \ \ cs_{i}=s_{i}c, \ \ s_ih_i = h_is_i=h_i. && \end{align*}
The submonoid $\mathcal{K}_n$ of $\mathcal{W}_n$ generated by $h_1,\dots,h_{n-1},c$ is called the Kauffman monoid. Identities in Kauffman monoids were studied in [1,2]. It has been shown that the identity checking problem for the monoids $\mathcal{K}_3$ and $\mathcal{K}_4$ is decidable in polynomial time.
Theorem. The identity checking problem for the monoid $\mathcal{W}_n$ with $n\ge4$ is coNP-complete.
The complexity of identity checking problem for the monoid $\mathcal{W}_3$ still remains unknown.