Monotone regular functions

A regular function is given by a deterministic 2way transducer: a deterministic 2way automaton with outputs. Over the word $w\in \Sigma^*$ such an automaton has an input tape with content ${\vdash} w {\dashv}$ and can move left and right in a read-only fashion (accepting runs have to end on $\dashv$). On every transition an output word in $\Gamma^*$ is written on a right-only output tape. Such a machine realizes a partial function $f:\Sigma^*\rightarrow\Gamma^*$, defined on words with an accepting run. Such a function $f$ is monotone if for any $u,v\in\Sigma^*$, if $u\leq v$ then $f(u)\leq f(v)$, where $\leq$ denotes the prefix order. A sequential 2way transducer is a syntactic restriction of deterministic transducers where no transition is allowed on symbol $\dashv$. Such a machine necessarily realizes a monotone function since the run over word $u$ is a prefix of the run over word $uv$. </p> Question: Can every monotone regular function be realized by a sequential 2way transducer?