Higher-dimensional barcode bound for Laplace eigenfunction combinations

Let $u\in C(\mathbb{R})$ be a non-negative function, and let $f$ be an $L^2$-normalized linear combination of eigenfunctions of $\Delta$ with eigenvalues $\lambda_i\leq\lambda$ on a Riemannian manifold $(M,g)$. Let $\Phi_u(f)$ be the barcode functional defined by summing the contributions determined by $u$ over all finite bars of $f$ in all degrees, and let $\|u\circ f\|$ denote the norm used in the source.

Higher-dimensional barcode-functional conjecture. There exists a constant $\kappa_g>0$ such that, for any $\lambda>0$,

$$\Phi_u(f)\leq\kappa_g(\lambda+1)^{\frac{n}{2}}\|u\circ f\|.$$

This conjecture extends the surface barcode estimates to arbitrary-dimensional Riemannian manifolds. The supplied material gives no evidence that it has been resolved.