We compute the lowest operator dimension Δ(Q) at large R-charge, Q, in 3D N=2 supersymmetric theory with a superpotential W=XYZ+τ*(X^3+Y^3+Z^3), as a function of the exact marginal coupling τ. The bahaviour of the dimension of such operators qualitatively differ depending on the theory has a moduli space of vacua (eg. at τ=0) or not, as this controls thee truncation of the chiral ring.
Based on an effective field theory approach of [Hellerman, Orlando, Reffert, MW] and combining it with the ε-expansion, we observe a crossover (not a phase transition), at around |τ|^2~1/Q, from Δ(Q, τ=0)~Q to Δ(Q)~Q^(3/2) as one moves away from τ=0. We will also comment on an unfinished attempt of not using the ε-expansion to work directly in 3D, using the constraint imposed by the duality group acting on the conformal manifold.