Gorenstein conjecture for the edge algebra of $\Gamma_{3k+2}$

Let $R_{\Gamma_{3k+2}}$ be the edge algebra associated with the graph $\Gamma_{3k+2}$. Gorenstein conjecture for $R_{\Gamma_{3k+2}}$. The algebra $R_{\Gamma_{3k+2}}$ is Gorenstein for every integer $k\geq 1$. The conjecture is motivated by the palindromic $h$-coefficients of its Hilbert series and computations for small $k$; the source gives no proof or resolution.