Regularity conjecture for horospherical tangent cones of symmetric spaces

Consider the horospherical tangent cones at infinity of symmetric spaces, with restricted root systems whose factors determine the relevant types. Regularity conjecture. The horospherical tangent cones at infinity of the symmetric spaces are irregular, except for the symmetric spaces whose restricted root system has all factors of type $A$. Explicit rank-two computations suggest this pattern, but the systematic determination of regularity is stated to be unknown.