SETH-based lower-bound conjecture for Induced Matching on graphs of bounded pathwidth

Let $(G,\ell)$ be an instance of \textsc{Induced Matching}, together with a path decomposition of $G$ of width $\operatorname{pw}$. The Strong Exponential Time Hypothesis (SETH) asserts that, for every $\epsilon>0$, no algorithm solves all such instances in time $(3-\epsilon)^{\operatorname{pw}}\cdot n^{\mathcal{O}(1)}$. SETH-based lower-bound conjecture. Unless the Strong Exponential Time Hypothesis is false, there does not exist a constant $\epsilon>0$ and an algorithm that, given an instance $(G,\ell)$ together with a path decomposition of $G$ of width $\operatorname{pw}$, solves \textsc{Induced Matching} in $(3-\epsilon)^{\operatorname{pw}}\cdot n^{\mathcal{O}(1)}$ time. This conjectured lower bound would establish that the base $3$ in parameterized algorithms for \textsc{Induced Matching} under pathwidth cannot be improved, assuming SETH.