Color class count function equivalence conjecture for partial colorings

Let $G_i'$ be the graph associated with a node $X_i$ of the smooth tree-decomposition, and let $f$ and $g$ be partial colorings of $G_i'$. For each tuple $\big( A^{(j)} \big)_{1 \le j \le 9}$ in $\mathcal{S}(X_i)$, let $CF_f\left(\big( A^{(j)} \big)_{1 \le j \le 9}\right)$ denote the number of color classes represented by the tuple, and let $CL_f\left(\big( A^{(j)} \big)_{1 \le j \le 9}\right)$ denote its color class function. Color class count function equivalence conjecture. If $f$ and $g$ have the same color class count function, namely

$$CF_f\left(\big( A^{(j)} \big)_{1 \le j \le 9}\right)=CF_g\left(\big( A^{(j)} \big)_{1 \le j \le 9}\right)$$

for every $\big( A^{(j)} \big)_{1 \le j \le 9}\in\mathcal{S}(X_i)$, then there is a permutation $\pi$ of the colors such that

$$CL_{\pi\circ f}\left(\big( A^{(j)} \big)_{1 \le j \le 9}\right)=CL_g\left(\big( A^{(j)} \big)_{1 \le j\le 9}\right)$$

for every $\big( A^{(j)} \big)_{1 \le j \le 9}\in\mathcal{S}(X_i)$. The conjecture would allow the algorithm to store color class count functions instead of full color class functions, potentially removing the restriction that the maximum degree $\Delta$ is constant and yielding a polynomial-time algorithm.