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.

[1] Chen Yuzhu, Hu Xun, Kitov N. V., Luo Yanfeng, Volkov M. V. Identities of the Kauffman monoid $\mathcal{K}_3$. Comm. Algebra 48:5 (2020), 1956-1968.
[2] Kitov N. V., Volkov M. V. Identities of the Kauffman monoid $\mathcal{K}_4$ and of the Jones monoid $\mathcal{J}_4$. In: Fields of Logic and Computation III [Lect. Notes Comp. Sci., volume 12180], Springer, Cham, 2020, 156-178.