Characterization of perfect graphs with regularity equal to induced matching number

Characterize all perfect graphs G for which the Castelnuovo–Mumford regularity of the extended double cover B_e(G) equals the upper irredundance number IR(G), equivalently, the induced matching number of B_e(G).

Background

The paper establishes equality between reg(B_e(G)) and α(G) for several graph classes contained in the perfect graphs, but explains that the equalities obtained earlier do not extend automatically to all perfect graphs.

By Proposition prop:im-double, the upper irredundance number IR(G) equals the induced matching number of B_e(G). The requested characterization therefore asks precisely when the regularity of the extended double cover coincides with its induced matching number among perfect graphs.