A conjectured upper bound for the iterated Mycielski L(2,1)-labeling number

Let $G$ be a graph of order $n\geq 1$ with maximum degree $\bigtriangleup$, and let $M^t(G)$ denote the $t$-fold iterated Mycielski graph of $G$, where $t\geq 1$. Let $\lambda(H)$ denote the $L(2,1)$-labeling number of a graph $H$.

Iterated Mycielski labeling conjecture. For all $t\geq 1$,

$$\lambda(M^t(G))\leq (2^t-1)(n+1)+\bigtriangleup^2.$$

This is proposed as a weaker form of the general $\bigtriangleup^2$-conjecture, using upper bounds for the $L(2,1)$-labeling number of Mycielski and iterated Mycielski graphs. Its resolution is not specified in the supplied text.