Reconfiguration conjecture for multipartite hypergraphs

Let $H$ be an $r$-partite $r$-graph, let $A$ be one of its $r$ partition classes, and let $\mathbf{RG}_{\mathrm{Mat}}(H,A;k)$ denote the matching reconfiguration graph on matchings of size $k$. Let $\tau(H)$ be the minimum vertex-cover size. Reconfiguration conjecture. For $r\geq 2$, if

$$k<\frac{\tau(H)}{r-1},$$

then

$$\mathbf{RG}_{\mathrm{Mat}}(H,A;k)$$

is connected. This extends the proved $r=3$ reconfiguration result in the spirit of the Henderson–Ryser conjecture. The claim is stated as a conjecture for all uniformities $r\geq 2$; no resolution is supplied in the source.