Monotonicity of distinguishing total chromatic number for the (8)-condition

Let $G$ be a simple graph and let $H$ be a proper subgraph of $G$. Write $\chi \, ”_{(8)}(G)$ for the minimum number of colors in an $(8)$-distinguishing total coloring. Monotonicity conjecture. No such pair satisfies

$$\chi \, ”_{(8)}(G)<\chi \, ”_{(8)}(H).$$

Equivalently, the proposed invariant would not increase when passing from a simple graph to a proper subgraph. The source contrasts this with known counterexamples for related strong and adjacent-strong edge and total invariants, but gives no resolution of the $(8)$-condition claim.