Equivariance conjecture for graded roots of delta-constant Gorenstein deformations

Let $(C_t,o)$ be a $\delta$-constant flat deformation of Gorenstein curves. The associated graded graph map

$$\operatorname{R}(C_{t\not=0},o)\to\operatorname{R}(C_{t=0},o)$$

is the map constructed from the deformation, and the graded roots carry their additional $\mathbb Z_2$-symmetries. Equivariance conjecture. The graded graph map is $\mathbb Z_2$-equivariant. This refines the deformation theory of graded roots; for general flat deformations of Gorenstein curves the corresponding equivariance can fail, so the delta-constant hypothesis is essential to the stated claim.