Splitting principle for objects of finite bosonic subdimension

Let $[?]$ be a 2-rig and let $r\triangleright [?]$ be an object of finite bosonic subdimension. Splitting principle conjecture. There exists a 2-rig $[?]$ and a map of 2-rigs $E\triangleright [?]$ such that $E(r)$ splits as a direct sum of finitely many bosonic subline objects, $E$ is faithful, conservative, and essentially injective, and the induced map

$$K(E)\colon K(\mathsf{R})\to K(\mathsf{R}')$$

is injective. The source presents this as a hoped-for generalization of a proved special case, but gives no resolution of the general statement.