Let $X$ be a smooth complex projective variety of dimension $n\geq2$, let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle over $X$ of rank $r\geq2$.

Does exists a Higgs subsheaf $\mathcal{K}$ of $\mathfrak{E}$ of rank $p$, for any $p\in\{1,\dotsc,r-1\}$?

For clarity: Higgs sheaf $\mathfrak{E}$ means a pair $(E,\varphi)$ where $E$ is a $\mathcal{O}_X$-coherent sheaf equipped with a morphism $\varphi\colon E\to E\otimes\Omega^1_X$ such that the composition $$ \varphi\wedge\varphi\colon E\xrightarrow{\varphi}E\otimes\Omega^1_X\xrightarrow{\varphi\otimes\operatorname{Id}}E\otimes\Omega^1_X\otimes\Omega^1_X\to E\otimes\Omega^2_X $$ vanishes. A Higgs subsheaf of a Higgs sheaf $(E,\varphi)$ is a $\varphi$-invariant subsheaf $G$ of $E$, i.e., $\varphi(G)\subset G\otimes \Omega_X^1$. A Higgs quotient of $\mathfrak{E}$ is a quotient of $E$ such that the corresponding kernel is $\varphi$-invariant. A Higgs bundle is a Higgs sheaf whose underlying coherent sheaf is locally free.

Let $X$ be a smooth complex projective variety of dimension $n\geq2$, let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle over $X$ of rank $r\geq2$.

Does exists a Higgs quotient sheaf $\mathcal{Q}$ of $\mathfrak{E}$ of rank $p$, for any $p\in\{1,\dotsc,r-1\}$?

For clarity: Higgs sheaf $\mathfrak{E}$ means a pair $(E,\varphi)$ where $E$ is a $\mathcal{O}_X$-coherent sheaf equipped with a morphism $\varphi\colon E\to E\otimes\Omega^1_X$ such that the composition $$ \varphi\wedge\varphi\colon E\xrightarrow{\varphi}E\otimes\Omega^1_X\xrightarrow{\varphi\otimes\operatorname{Id}}E\otimes\Omega^1_X\otimes\Omega^1_X\to E\otimes\Omega^2_X $$ vanishes. A Higgs subsheaf of a Higgs sheaf $(E,\varphi)$ is a $\varphi$-invariant subsheaf $G$ of $E$, i.e., $\varphi(G)\subset G\otimes \Omega_X^1$. A Higgs quotient of $\mathfrak{E}$ is a quotient of $E$ such that the corresponding kernel is $\varphi$-invariant. A Higgs bundle is a Higgs sheaf whose underlying coherent sheaf is locally free.