ISI competitive equilibrium allocation doubt

[Asd1995]

Ref. Q.29, we are asked what are the possible competitive equilibrium allocations.

Since both utility functions (2x plus y and x plus 2y) are convex, shouldn't all pareto efficient allocations be competitive equilibrium for some ratio of prices? Or does this hold when preferences are strictly convex?

Even if they are strictly convex, should we be checking for corner points individually anyway?

[Daffodils]

A competitve eqm is PE but all PE points are not CE -- FIRST WELFARE THRM.

[Asd1995]

If preferences are convex every CE is PE and every PE IS CE for some set of prices.