The realization conjecture for winding-number invariants of almost embeddings of $K_4$

Let $K_4$ be the complete graph on vertices $1,2,3,4$, and let $f:K_4\to\mathbb R^2$ be an almost embedding. For an oriented cycle $C$ and a vertex $v$, write $w_f(C,v)$ for the winding number of the image of $C$ about $f(v)$. For $j\in[4]$, set

$$\begin{aligned} w_f(C_1,1)&=w_f(234,1),\\ w_f(C_2,2)&=w_f(134,2),\\ w_f(C_3,3)&=w_f(124,3),\\ w_f(C_4,4)&=w_f(123,4). \end{aligned}$$

Realization conjecture. For any integers $n_1,n_2,n_3,n_4$ whose sum is odd, there exists an almost embedding $f:K_4\to\mathbb R^2$ such that

$$w_f(234,1)=n_1,\quad w_f(134,2)=n_2,\quad w_f(124,3)=n_3,\quad w_f(123,4)=n_4.$$

This asserts that the parity condition supplied by the theorem that the corresponding signed sum is odd is the only restriction on these four winding numbers. The supplied text gives examples and references, but does not state whether the realization claim has been proved; its status is therefore left open.