Self-duality conjecture for finite Grassmann algebras

Let ${\mathfrak{B}}_{{L}}$ be the Grassmann algebra with $L$ generators, and write ${\mathfrak{B}}_{{L},\mathrm{ev}}$ and ${\mathfrak{B}}_{{L},\mathrm{od}}$ for its even and odd parts. Let ${\bf{L}}_{{\mathfrak{B}}_{{L},\mathrm{ev}}}({\mathfrak{B}}_{{L},\mathrm{od}}:{\mathfrak{B}}_{{L}})$ denote the set of continuous ${\mathfrak{B}}_{{L},\mathrm{ev}}$-linear maps from ${\mathfrak{B}}_{{L},\mathrm{od}}$ to ${\mathfrak{B}}_{{L}}$, meaning that $f(\lambda X)=\lambda f(X)$ for $\lambda\in {\mathfrak{B}}_{{L},\mathrm{ev}}$ and $X\in {\mathfrak{B}}_{{L},\mathrm{od}}$.

Self-duality conjecture. For every $f\in {\bf{L}}_{{\mathfrak{B}}_{{L},\mathrm{ev}}}({\mathfrak{B}}_{{L},\mathrm{od}}:{\mathfrak{B}}_{{L}})$, there exists $u_f\in {\mathfrak{B}}_{{L}}$ such that

$$f(X)=X\!\cdot u_f$$

for every $X\in {\mathfrak{B}}_{{L},\mathrm{od}}$.

The claim is motivated by the expected self-duality of the odd part of a Grassmann algebra, but the paper states that Kazuo Masuda supplied a counterexample when $L=2$: the map defined by $f(X_1\sigma_1+X_2\sigma_2)=X_1\sigma_2$ is continuous and ${\mathfrak{B}}_{2,\mathrm{ev}}$-linear, yet no such $u_f$ exists. Thus the conjecture is refuted for finite $L$, while the paper suggests that self-duality may require countably many Grassmann generators.