Chromatic boundedness for graphs with spaghetti tree- and path-decompositions

A spaghetti tree-decomposition of a graph $G$ is a tree-decomposition $(T,\{B_t\}_{t\in V(T)})$ rooted at some vertex, such that the bags containing each vertex of $G$ induce a directed path when the edges of $T$ are oriented away from the root. A path-decomposition of $G$ is a tree-decomposition whose underlying tree is a path. Let $\chi(G)$ denote the chromatic number of $G$.

Chromatic boundedness conjecture. There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that $\chi(G)\leq f(k)$ for every $k\geq 1$ and every graph $G$ admitting a spaghetti tree-decomposition $(T,\{B_t\}_{t\in V(T)})$ and a path-decomposition $(P,\{B_p\}_{p\in V(P)})$ satisfying

$$|B_t\cap B_p|\leq k\qquad\text{for every }t\in V(T)\text{ and }p\in V(P).$$

This asks whether the chromatic number is bounded solely in terms of the maximum intersection size between the bags of the two decompositions, extending the use of tree-decomposition methods in bounding chromatic number. The source presents it as an open problem and gives no resolution.