Finite-quiver socle quotient conjecture for Leavitt path algebras

Let $Q$ be a finite quiver, let $\widehat{Q}$ denote its Kronecker square, and let $L_{\mathbb K}(Q)$ be the Leavitt path algebra over a field $\mathbb K$. Let $\operatorname{Soc}(L_{\mathbb K}(\widehat{Q}))$ be its socle, let $\overline{\mathop{\rm P}_l(Q)}$ be the hereditary saturated closure of the point-line vertices of $Q$, and let $Q\setminus\overline{\mathop{\rm P}_l(Q)}$ be the quiver obtained by removing that closure. Finite-quiver socle quotient conjecture. There exists a short exact sequence

$$\operatorname{Soc}(L_{\mathbb K}(\widehat{Q}))\hookrightarrow L_{\mathbb K}(\widehat{Q})\twoheadrightarrow L_{\mathbb K}(Q\setminus\overline{\mathop{\rm P}_l(Q)})\otimes_G L_{\mathbb K}(Q\setminus\overline{\mathop{\rm P}_l(Q)}),$$

or, equivalently,

$$\frac{L_{\mathbb K}(\widehat{Q})}{\operatorname{Soc}(L_{\mathbb K}(\widehat{Q}))}\cong L_{\mathbb K}(Q\setminus\overline{\mathop{\rm P}_l(Q)})\otimes_G L_{\mathbb K}(Q\setminus\overline{\mathop{\rm P}_l(Q)}).$$

For finite quivers the transfinite socle collapses to the ordinary socle, yielding this finite form of the preceding conjecture.