The proper containment conjecture for identities of principal subalgebras

Let $Q$ be a finite $PI$ but non-acyclic quiver, and let $\mathbb{K}Q_{\operatorname{\geq 1}}$ denote its principal subalgebra generated by paths of positive length. Suppose there exists a vertex $i\in Q_0$ with no non-trivial path $p$ satisfying $s(p)=t(p)=i$.

Proper containment conjecture. The ideal of polynomial identities of the path algebra is properly contained in that of its principal subalgebra:

$$\operatorname{Id}(\mathbb{K}Q)\subsetneq\operatorname{Id}(\mathbb{K}Q_{\geq 1}).$$

The conjecture proposes a general explanation for the examples in which the two identity ideals differ for finite $PI$ but non-acyclic quivers. Its status is not established by the supplied text.