It is well known the following result:
Lemma: Let $F\rightarrow E\rightarrow B$ be a fibration with $B$ connected and simply connected. Suppose that $F$ is $n$-acyclic, i.e. $H^{p}\left( F;% %TCIMACRO{\U{211a} } %BeginExpansion \mathbb{Q} %EndExpansion \right) =H^{p}\left( pt;% %TCIMACRO{\U{211a} } %BeginExpansion \mathbb{Q} %EndExpansion \right) $ for $p\leq n$. Then $$ H^{p}\left( B; \mathbb{Q} \right) \rightarrow H^{p}\left( E;% %TCIMACRO{\U{211a} } %BeginExpansion \mathbb{Q} %EndExpansion \right) $$ is an isomorphism for $p\leq n$ and a monomorphism for $q=n+1$.
The converse of this lemma is true? That is, if $$ H^{p}\left( B;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \rightarrow H^{p}\left( E;% %TCIMACRO{\U{211a} } %BeginExpansion \mathbb{Q} %EndExpansion \right) $$ is an isomorphism for $p\leq n$ and a monomorphism for $q=n+1$, then $F$ is $% n$-acyclic?