Makhlouf–Silvestrov conjecture for infinitesimal deformations of $\mathfrak{sl}_2$

Let $(V,[\cdot,\cdot]_0)\cong\mathfrak{sl}_2(\mathbb K)$ and set $\alpha_0=\mathrm{id}_V$. Suppose that

$$[\cdot,\cdot]_t=[\cdot,\cdot]_0+t[\cdot,\cdot]_1,\qquad \alpha_t=\alpha_0+t\alpha_1$$

define an infinitesimal Hom--Lie deformation over $\mathbb K[t]/(t^2)$, meaning that $(V\otimes_{\mathbb K}\mathbb K[t]/(t^2),[\cdot,\cdot]_t,\alpha_t)$ is a Hom--Lie algebra. Assume in addition that $(V,[\cdot,\cdot]_0,\alpha_1)$ is a Hom--Lie algebra.

Makhlouf--Silvestrov conjecture. The resulting Hom--Lie algebra $(V,[\cdot,\cdot]_t,\alpha_t)$ is in fact a Lie algebra: the bracket $[\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity.

This conjecture concerns whether the additional Hom--Lie condition on the first-order twisting rules out genuinely non-Lie infinitesimal deformations of $\mathfrak{sl}_2(\mathbb K)$. It was formulated from computer-algebra evidence, which found only examples whose brackets satisfied the ordinary Jacobi identity; its resolution is not established in the supplied source.