The nullhomotopy-dependence conjecture for Floer-theoretic nearby Lagrangian torsion

Let $L\subset T^*S^{4k}$ be a Lagrangian homotopy sphere, and fix two nullhomotopies $\gamma_t^0$ and $\gamma_t^1$ of its stable Gauss map $L\to U/O$. Let $\tau_{2k}(L,\gamma_t^i)\in H^{4k}(S^{4k};\mathbf{Z})$ be the corresponding cohomology classes. Define $E\to S^{4k}$ as the stable real vector bundle classified by the composite map $S^{4k}\to BO$ formed from a homotopy inverse to the projection $L\to S^{4k}$, the map $L\to\Omega(U/O)$ obtained by concatenating $\gamma_{1-t}^0$ with $\gamma_t^1$, the Bott periodicity map $\Omega(U/O)\simeq\mathbf{Z}\times BO$, and the map $\mathbf{Z}\times BO\to BO$.

Nullhomotopy-dependence conjecture. The classes $\tau_{2k}(L,\gamma_t^i)$, for $i=1,2$, differ by $p_k(E)$.

If true, this would make the Floer-theoretic nearby Lagrangian torsion well-defined as a coset modulo the image of $p_k$, yielding an invariant in $\mathbf{R}/\lambda_k\mathbf{Z}$. The source presents this as a conjecture and gives no evidence of a resolution.