Iwase–Sakai conjecture on monoidal topological complexity

Let $X$ be a locally finite simplicial complex, and let $\operatorname{TC}(X)$ denote its topological complexity and $\operatorname{TC}^M(X)$ its monoidal topological complexity. The latter is the relative version defined using the diagonal map $\Delta:X\rightarrow X\times X$. I-S conjecture. For every locally finite simplicial complex $X$,

$$\operatorname{TC}(X)=\operatorname{TC}^M(X).$$

Iwase and Sakai proved that, for locally finite simplicial complexes (more generally, Euclidean Neighborhood Retracts), the two quantities differ by at most one. The equality was still described in the paper as an open problem, and the paper reports results supporting it.