Regularity formula for t-path ideals of unicyclic graphs

Let $G$ be a connected simple graph having at most one cycle $C$ of length $m$. Let $t\geq 2$ be a natural number. Write $S$ for the polynomial ring associated with $G$, $I_t(G)$ for its $t$-path ideal, $\nu_t(G)$ for the maximum number of pairwise suitable $t$-paths in $G$, and call $G$ $t$-proximal when it satisfies the corresponding $t$-proximity condition. Then

$$\operatorname{reg}(S/I_t(G))=\begin{cases} (t-1)\nu_t(G)+\left(\operatorname{reg}(S/I_t(C))-(t-1)\nu_3(C)\right), & \text{if }G\text{ is }t\text{-proximal},\\ (t-1)\nu_t(G), & \text{otherwise}. \end{cases}$$

Regularity conjecture for t-path ideals. The displayed formula should hold. In particular,

$$\operatorname{reg}(S/I_t(G))=(t-1)\nu_t(G)$$

when $G$ is a tree. This conjecture is presented as a natural generalization of the paper's theorem for $3$-path ideals; the supplied text gives no resolution status for the general $t$-path statement.