A conjectural formula for $\lambda_g\mathrm{DR}_g(a,-a)$ is true in Gorenstein quotient

In this note we prove that a conjectural formula for the class $\lambda_g \mathrm{DR}_g(a,-a)\in R^{2g}(\overline{\mathcal{M}}_{g,2})$ proposed recently by Buryak-Iglesias-Shadrin is true in the Gorenstein quotient of the ring $R^*(\overline{\mathcal{M}}_{g,2})$. As a corollary we prove the strong $\mathrm{DR} \backslash \mathrm{DZ}$ equivalence conjecture for one-point correlators.