The barcode resolution conjecture for compactly supported piecewise linear sheaves

Let $\mathbb{V}$ be the ambient real vector space, let $\gamma$ be the cone defining the sheaf-theoretic setting, and let $F\in\operatorname{Der}_{\mathrm{PL},\gamma^{\circ a}}({\bf k}_{\mathbb{V}})$ have compact support. A barcode $\gamma$-sheaf is an object in the essential image of the fully faithful functor from the category of $\gamma$-barcodes to the relevant derived category, hence is a finite direct sum of sheaves associated with barcode pieces.

Barcode resolution conjecture. There exists a bounded complex

$$F^{\,\bullet}\in {\rm C}^{\mathrm{b}}\bigl(\operatorname{Mod}({\bf k}_{\mathbb{V}_\gamma})\bigr)$$

whose image in $\operatorname{Der}_{\mathrm{PL},\gamma^{\circ a}}({\bf k}_{\mathbb{V}})$ is isomorphic to $F$, and such that every component $F^j$ of $F^{\,\bullet}$ is a compactly supported barcode $\gamma$-sheaf.

This conjecture predicts that every compactly supported piecewise linear $\gamma$-sheaf can be represented by a bounded complex built from barcode $\gamma$-sheaves. It concerns the failure of the barcode functor to be essentially surjective in dimensions greater than one by asserting essential surjectivity after passage to bounded complexes; the supplied source gives no resolution status.