Non-loose realizations from non-zero hat Legendrian invariants

Let $L\hookrightarrow Y$ be an $n$-component link admitting a Legendrian realization $\mathcal L$ in $(Y,\xi)$ with non-zero $\operatorname{\widehat{\mathfrak{L}}}(\mathcal L)$. Then there is a non-loose realization $\mathcal M$ of $L$ in an overtwisted structure $\eta$ on $Y$ such that

$$\mathfrak t_\eta=\mathfrak t_\xi\in\operatorname{Spin^c}(Y)$$

and

$$d_3(\eta)=d_3(\xi)-2A\bigl(\operatorname{\widehat{\mathfrak{L}}}(\mathcal L)\bigr)+n.$$

Non-loose realization conjecture. If the hat Legendrian invariant of a Legendrian realization is non-zero, then the construction produces a non-loose realization in an overtwisted contact structure with the stated $\operatorname{Spin^c}$ and $d_3$ invariants. The preceding discussion notes that the corresponding case for taut foliations is confirmed, while the general assertion is presented as an expectation and remains open.