What is the status of this strong form of Hedetniemi's conjecture?

Question

In this question, a graph is a finite, undirected graph without loops or multiple edges, and a colouring of a graph is a proper vertex colouring. The product $G \times H$ of graphs $G$ and $H$ is the graph whose vertex-set is the product of the vertex-sets of $G$ and $H$ and whose edge-set is the product of the edge-sets of $G$ and $H$, with the obvious incidence relation.

Let $G$ and $H$ be graphs. Any $n$-colouring of $G$ gives rise to an $n$-colouring of $G \times H$: just paint $(x, y)$ the same colour as $x$. (Or, if you prefer, an $n$-colouring of a graph is just a homomorphism into the complete graph $K_n$, so we can compose the colouring $G \to K_n$ with the projection $G \times H \to G$ to obtain a colouring $G \times H \to K_n$.) Similarly, any $n$-colouring of $H$ gives rise to an $n$-colouring of $G \times H$. Let us say that a colouring of $G \times H$ arising in one of these two ways is obtained by projection.

The previous paragraph makes it clear that $\chi(G \times H) \leq \min\{ \chi(G), \chi(H) \}$, where $\chi$ means chromatic number. Hedetniemi's conjecture states that this is an equality. In other words, it says that there are no colourings of a product more economical than those obtained by projection. My question:

Let $G$ and $H$ be graphs. Is every colouring of $G \times H$ with $\chi(G \times H)$ colours obtained by projection?

The answer can't be known to be yes, unless I've missed some news, since that would imply Hedetniemi's conjecture. But perhaps the answer is no, or perhaps it's known that this apparently stronger conjecture would actually be implied by Hedetniemi's original conjecture.

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Edit  Assume the graphs are connected, otherwise the answer is trivially no. (E.g. consider $(K_2 \sqcup K_2) \times K_2$.)

Answer 1

The answer is no. In the comments, Gil Kalai suggested looking at products of two odd cycles, and indeed, this yields a counterexample.

Consider $C_3 \times C_5$, where $C_n$ denotes the $n$-cycle. This has chromatic number $3$, and there is a 3-colouring given by $$ \begin{pmatrix} 1&3&1&2&3\\ 1&2&1&2&3\\ 1&2&1&2&3 \end{pmatrix} $$ (in what I hope is obvious notation). This is plainly not obtained by projection.

In fact, this is also a counterexample to a weaker form of this strong form of Hedetniemi's conjecture: that every colouring of $G \times H$ by $\chi(G \times H)$ colours is obtained by projection after some automorphism of $G \times H$. For any such colouring of $C_3 \times C_5$ has either 5 of each colour or 3 of one and 6 of each of the other two; but neither is the case for the colouring shown above.