Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity

We show that whenever $$[\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad \alpha_t = \mathrm{id} + t\alpha_1$$ define an infinitesimal Hom--Lie deformation of $\mathfrak{sl}_2(\mathbb K)$ over $\mathbb K[t]/(t^2)$ and $(\mathfrak{sl}_2(\mathbb K),[\,\cdot,\cdot]_0,\alpha_1)$ is a Hom--Lie algebra, then the deformed bracket $[\,\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity over $\mathbb K[t]$. This solves a conjecture of Makhlouf and Silvestrov from 2010.