Nonsingularity conjecture for elliptic inflectionary curves

Let $k\geq 1$, and let $\overline{\mathcal{C}(1,k)}$ be the projective plane inflectionary curve defined by the homogenized inflection polynomial $P_{1,k}$, with singular points $p_1=[0:0:1]$, $p_2=[0:1:0]$, and $p_3=[1:1:1]$. Nonsingularity conjecture. For every positive integer $k\geq 1$, the plane curve $\overline{\mathcal{C}(1,k)}$ is nonsingular along $\mathbb{P}^2\setminus\{p_1,p_2,p_3\}$. The preceding symmetries imply that the singularities at the three listed points are analytically isomorphic; the conjecture concerns the absence of any other singularities.