Vertex-deletion conjecture for the upper orientable total domination number

Let $G$ be a graph in $\mathcal{C}$ and let $v\in V(G)$ be such that $G-v\in\mathcal{C}$. Vertex-deletion conjecture.

$${\rm DOM}_t(G)\leq {\rm DOM}_t(G-v)+1.$$

The conjecture proposes the missing lower bound from the cited vertex-deletion result; the corresponding upper bound for induced subgraphs is known, but this inequality was not proved in the paper.