Stability of equivariant logarithmic tangent sheaves on toric varieties of Picard rank two

For an equivariant log pair (X,D) where X is a normal toric variety and D a reduced Weil divisor, we study slope-stability of the logarithmic tangent sheaf 𝒯 X (-logD). We give a complete description of divisors D and polarizations L such that 𝒯 X (-logD) is (semi)stable with respect to L when X has a Picard rank one or two.