Existence and zigzag bound conjecture for tight polyhedra of type $4_n$

Let $4_n$ be a graph of type $4_n$, and call it z-knotted when it has one zigzag up to reversal and tight when it satisfies the source's tightness condition. Let $n$ denote the number of vertices.

$4_n$ existence and zigzag conjecture. (i) A z-knotted $4_n$ exists if and only if $n\geq 30$ and $n\equiv 2\pmod 4$. (ii) A tight $4_n$ exists if and only if $n\geq 8$ and $n\ne 10,14$. (iii) Every tight $4_n$ has at most eight zigzags.

These assertions give existence criteria for z-knotted and tight graphs of type $4_n$, together with a uniform upper bound on the number of zigzags; the supplied text does not provide a resolution beyond presenting them as conjectural claims.