Simultaneous existence of two ultrafilters with strong generating sequences of different lengths

Establish whether it is consistent that there exist two ultrafilters U and W on a fixed cardinal κ such that U admits a strong generating sequence of length κ^+ and W admits a strong generating sequence of length κ^{+++}, yielding Sp_T(U·W)=Sp_T(U)∪Sp_T(W) with non-interval behavior.

Background

The authors investigate the possible shapes of the Tukey point spectrum Sp_T(U) of an ultrafilter U and raise whether it can fail to be an interval of regular cardinals. A natural approach would be to combine two ultrafilters with strong generating sequences of distinct regular lengths, e.g., κ^+ and κ^{+++}, to force Sp_T(U·W)=Sp_T(U)∪Sp_T(W)={κ^+, κ^{+++}}.

However, they explicitly state that it is unknown if two such ultrafilters can coexist simultaneously in the same model, which would be necessary to realize this phenomenon.