In 1990, Deodhar [3] showed that the non-negativity of Kazdan-Lusztig polynomials implies the expansion of any monomial of (dual) Kazhdan-Lusztig basis elements $C'_{s_{i_1}}\cdots C'_{s_{i_r}}=\sum_x L_xC'_x$ with $s_{i_j}\in S$ relative to the (dual) Kazhdan-Lusztig basis $\{C'_w:w\in W\}$ of the Hecke algebra of the Coxeter system $(W,S)$ has non-negative coefficients, i.e. that $L_x\in\mathbb N[q^{\frac12}+q^{-\frac12}]$. In fact, though he does not state it, Deodhar's work implies that the non-negativity conjecture is equivalent to these monomial expansions having non-negative $L_x$.

Now, given that cases of non-negativity conjecture have been settled by interpreting the coefficients of the Kazhdan-Lusztig polynomials in terms of intersection cohomology, do the coefficients of $L_x$ also have such an interpretation in those cases?

EDIT: As Alexader Woo pointed out, the answer is yes and one reference is Springer [2], who actually shows that $C'_w\cdot C'_x=\sum_y L_{w,x;y}C'_y$ with the coefficients of $L_{w,x;y}\in\mathbb N[q^{\frac12},q^{-\frac12}]$ coming from intersection cohomology. This motivates me to change my question to whether non-negativity of the Kazhdan-Lusztig polynomials alone implies that all products of dual Kazhdan-Lusztig basis elements lie in the cone spanned by linear combinations of dual Kazhdan-Lusztig basis elements with non-negative coefficients.

Relatedly perhaps, I should ask for clarification on the relationship between the Hecke algebra and the quantized enveloping algebra equipped with a canonical basis consisting of isomorphism classes of certain perverse sheaves that Lusztig associated to any graph in 1993 [1]. As far as I understand (which is not far at all), the two algebras have formally similar properties, but in the latter non-negativity comes for free since any monomial of special basis elements $F_i^{(a)}$ has an expansion $F_{i_1}^{(a_1)}\cdots F_{i_m}^{(a_m)}=\sum_{b\in B;d\in\mathbb Z}M(b,i,a,d)v^{d+d_0}b$, where $M(b,i,a,d)$ is the number of times the shifted pervese sheaf b[d] appears in a direct sum decomposition of the direct image of a constant sheaf under a proper morphism depending only on the sequences $i$ and $a$, and $d_0$ is likewise a integer depending only on $i$ and $a$.

[1]: G. Lusztig. Tight monomials in quantized enveloping algebras. *Quantum deformations of algebras and their representations. ed. A.Joseph et al., Isr.Math.Conf.Proc.7(1993), Amer.Math.Soc. 117-132.

[2]: T.A. Springer. Quelques applications de la cohomologie d'intersection. *Asterisque*, tome 92-93 (1982), Séminaire Bourbaki. p. 249-273.

[3]: Vinay V. Deodhar. A Combinatorial Setting for Questions in Kazhdan-Lusztig Theory. *Geometriae Dedicata*, 36(1):95-119, 1990.