Increasing on a uniform above-above window

Fix a type $R$ equipped with a linear order that is dense and has no maximum and no minimum, let $M$ be an o-minimal structure on $R$, let $I$ and $B$ be sets of unary tuples over $R$, let $f$ be an $M$-definable function from $I$ to $B$, and let $a$ and $b$ be endpoints over $R$; assume $f$ is injective in the sense that for any unary tuples $x,y$ belonging to $I$, equality of the $f$-values at $x$ and $y$ implies $x=y$, assume every unary tuple lying in the open interval $(a,b)$ belongs to $I$, assume the left endpoint $a$ is strictly below the right endpoint $b$, and assume the following uniform two-sided local lower-bound property: for every $x$ belonging to $I$ and also to $(a,b)$ there exist unary tuples $c_1,c_2$ both in $(a,b)$ with $c_1<x<c_2$ in the strict order on unary tuples such that, first, for every $y$ belonging to $I$ and to $(a,b)$ with $c_1<y<x$ the scalar component of the $f$-value at $x$ is strictly less than the scalar component of the $f$-value at $y$, and second, for every $y$ belonging to $I$ and to $(a,b)$ with $x<y<c_2$ the scalar component of the $f$-value at $x$ is strictly less than the scalar component of the $f$-value at $y$; then there exist endpoints $a_2$ and $b_2$ with $a_2$ strictly below $b_2$ such that every unary tuple in the open interval $(a_2,b_2)$ belongs to $I$ and $f$ is strictly increasing on $I\cap (a_2,b_2)$ in the sense that for any $x,y$ belonging to $I$ and to $(a_2,b_2)$ with $x$ strictly below $y$, the scalar component of the $f$-value at $x$ is strictly below that at $y$, where the universal claims are vacuously true if the relevant intersections $I\cap(a,b)$ or $I\cap(a_2,b_2)$ contain zero or one point, while the hypotheses that $(a,b)$ is nondegenerate and contained in $I$ normally force $I\cap(a,b)$ to be nonempty in a dense order without ends.