Let $X$ be a connected simplicial set. If $X$ is an Kan complex and $k\geq 0$, then every element $$ \tilde f\in\operatorname{Hom}_{Ho(sSet)}(S^k,X) $$ of the homotopy classes from $S^k$ to $X$ lifts to a map $f\colon S^k\to X$ of simplicial sets. This is an abstract fact about model categories which may be applied since every simplicial set (and in particular $S^k$) is cofibrant and $X$ is fibrant.
Suppose $X$ satisfies the Kan condition only ''from a certain fixed $N$ on'', i.e. for all $n\geq N$, every diagram $$ \begin{array}{rcl} \Lambda^n_j &\to &X\\ \downarrow &&\\ \Delta^n \end{array} $$ obtains a lift $l\colon \Delta^n\to X$ (This is the usual Kan condition for $N=0$).
Does there exist an $k>N$ such that every element $\tilde f$ in $\operatorname{Hom}_{Ho(sSet)}(S^k,X)$ can be lifted to $f\colon S^k\to X$, if $X$ satisfies the weaker condition above?
(I think that there exists a counterexample but I cannot think of one...)