We study the natural resolution of the conjugated Haar multiplier
$M_{w^{\frac{1}{2}}}T_{\sigma}M_{w^{-\frac{1}{2}}},$ where the multiplication
operators $M_{w^{\pm\frac{1}{2}}}$ are decomposed into their canonical
paraproduct decompositions. We prove that each constituent operator obtained
from this resolution has a linear bound on $L^2(\mathbb{R}^d;w)$ in terms of
the $A_{2}$ characteristic of $w$. The main tools used are a product formula
for Haar coefficients, the Carleson Embedding Theorem, the linear bound for the
square function, and the well-known linear bound of $T_{\sigma}$ on
$L^2(\mathbb{R}^d,w).$