The inverted Habiro-series formula for knot-complement invariants

Let $K$ be a knot with Alexander polynomial $\Delta_K\neq1$, let $F_K(x,\tau)$ be its knot-complement invariant, and let $a_{-m}(K;q)$ denote inverted Habiro coefficients. Inverted Habiro-series formula conjecture. When the inverted Habiro-series expressions yield the correct $\pm p$ surgeries on a plumbed knot, inserting a defect of highest weight $\nu\vec\omega$ gives

$$\hat Z_b(S_{-p}^3(K);W_\nu;\tau)=\epsilon q^{d-b(2p-b)/(4p)}\sum_{n=0}^{\infty}a_{-n-1}(K;q)\sum_{\ell=1}^n\lambda_\ell^{n,\nu}(q)\frac{q^{\ell^2}}{(q^{\ell+1};q)_\ell}P_\ell^{p,b+\nu}(q^{-1})+p(\tau),$$

when $b\equiv\nu\pmod 2$, and it is zero otherwise; similarly,

$$\hat Z_b(S_{+p}^3(K);W_\nu;\tau)=\epsilon q^{-d+b(2p-b)/(4p)}\sum_{n=0}^{\infty}a_{-n-1}(K;q)\sum_{\ell=1}^n\lambda_\ell^{n,\nu}(q)\frac{(-1)^\ell q^{\ell(\ell+1)/2}}{(q^{\ell+1};q)_\ell}P_\ell^{p,b+\nu}(q)+p(-\tau),$$

under the same parity condition, with $p(\tau)$ finite. Here the finite polynomials $\lambda_\ell^{n,\nu}(q)=\lambda_\ell^{n,\nu}(q^{-1})$ are defined by the two displayed rational-function decompositions in the source. The claim extends the inverted Habiro-series surgery formulas to defect operators; it is conditional on the underlying surgery expressions being correct, and no resolution is given.