On the impossibility of parabolic factorization of certain Kazhdan-Lusztig basis elements

The Kazhdan-Lusztig basis \$\{C'_w \mid w \in S_n\}\$ of the Hecke algebra \$H_n\$ is related to the natural basis \$\{T_v \mid v \in S_n\}\$ of \$H_n\$ by a matrix whose entries are recursively-defined polynomials \$\{P_{v,w}(q) \mid v,w \in S_n\}\$ in \$\mathbf N[q]\$ known as the Kazhdan-Lusztig polynomials. No known combinatorial formula interprets the coefficients of these polynomials as set cardinalities.  Nevertheless, some results which depend upon pattern avoidance in the permutation \$w\$ permit one to factor the Kazhdan-Lusztig basis element \$C'_w\$ in a way which provides combinatorial formulas for coefficients of the polynomials \$\{P_{v,w}(q) \mid v \in S_n\}\$ having second index \$w\$.  No characterization of the permutations \$w\$ permitting such a factorization is known. We present a negative result: conditions on \$w\$ which imply that such a factorization does not exist.