The planar Linek conjecture for independent sets

Let $i(P)$ denote the number of independent sets of a connected planar graph $P$. Planar Linek's Problem. Every positive integer is equal to $i(P)$ for some connected planar graph $P$. The conjecture strengthens the proved result that almost every positive integer is realized by a connected planar graph. Computation supports it by realizing the $823$ values missing from the cited tree computation, but the assertion for every positive integer remains open.