Leander's generalized Helleseth conjecture for permutations

Let $L$ be a finite field, let $L^\times$ be its multiplicative group, and let $f\colon L\to L$ be a permutation. For each $\lambda\in L^\times$, write $(\lambda f)(x)=\lambda f(x)$ and define

$$\mathfrak D(\lambda f)=\prod_{a\in L^\times}\widehat{\lambda f}(a),$$

where

$$\widehat{\lambda f}(a)=\sum_{x\in L}\mu(ax+\lambda f(x)).$$

Leander's generalized Helleseth conjecture. If $f$ is a permutation of $L$, then

$$\prod_{\lambda\in L^\times}\mathfrak D(\lambda f)=0.$$

The source presents this as a possible generalization proposed by Leander, motivated by the failure of Helleseth's conjecture to extend to all permutations; the supplied text gives no resolution.