Does a problem remain tractable If a single discrete variable becomes continuous?

Question

Let $\mathcal{F}$ be a family of pairs of the form $(A,b)$, where $A$ is an integer matrix and $b$ is an integer vector with the same number of rows. For every integer $k$, define $L(\mathcal{F}, k)$ to be the following family of decision problems.

INPUT: a pair $(A,b)\in \mathcal{F}$.

OUTPUT: "True" if there exists a real vector $x \geq 0$ satisfying $A x \leq b$, where for every $i > k$, $x_i$ must be an integer (for $i\leq k$, $x_i$ can be any real number).

If $k=0$, then these are simply integer programming problems, and $L(\mathcal{F}, k)$ is NP-hard in general. But it is in P for some subclasses of matrices and vectors, for example, when for all $(A,b)\in \mathcal{F}$, the matrix $A$ is of totally unimodular.

I am interested in the effect of $k$ on the complexity of $L(\mathcal{F}, k)$. In particular, if $L(\mathcal{F}, k)$ is in P for some $k$, does it imply that $L(\mathcal{F}, k+1)$ is in P too?

Stated informally: if we take a single discrete variable and allow it to be continuous, does the problem remain tractable?

EDIT: Here is an example showing that the opposite is possible: if we start from an NP-hard problem, and convert a single discrete variable to continuous, the problem may become tractable. Consider the following family of linear programs: $$ a_1 x_1 + \cdots + a_n x_n = (a_1+ \cdots +a_n)/2 \\ 0 \leq x_i \leq 1 ~~~ \forall i\in\{1,\ldots,n\} \\ x_i \in \mathbb{Z} ~~~ \forall i\in\{1,\ldots,n\} $$ where the input is the integer vector $a_1,\ldots,a_n$ and the output is "true" if there is a vector $x_1,\ldots,x_n$ satisfying the constraint. This LP represents the NP-hard partition problem.

Suppose w.l.o.g. that the input arrives in descending order ($a_1\geq \cdots \geq a_n$). Then, if we allow a single variable $x_1$ to be continuous (we remove the constraint $x_1\in \mathbb{Z}$), then the problem becomes tractable: if the largest item can be split, then the answer is always "yes". So, $L(\mathcal{F}, 1)$ is in $P$ while $L(\mathcal{F}, 0)$ is NP-hard.

Is there an $\mathcal{F}$ for which the opposite holds --- $L(\mathcal{F}, 1)$ is in $P$ while $L(\mathcal{F}, 0)$ is NP-hard?

EDIT 2: Note that, without the restriction to linear problems, even making all variables continuous might make a problem harder. See this answer.

Answer 1

If a linear program has some integer and continuous variables, then the problem is known as Mixed-Integer Linear Program. Moreover, if the number of integer variables are constant, then the problem can be solved in polynomial time (see Proposition 8.1 from here).

Therefore, if any $L(\mathcal{A}, \mathcal{B}, k)$ is in $\mathsf{P}$ for some $k$, and suppose that $n-k = O(1)$ where $n$ is the number of elements in $x$. Then, it trivially implies that $L(\mathcal{A}, \mathcal{B}, k+1)$ is in $\mathsf{P}$ too, since $L(\mathcal{A}, \mathcal{B}, k+1)$ can be solved in polynomial time (there are at most $n-k-1 = O(1)$ integer variables in $L(\mathcal{A}, \mathcal{B}, k+1)$).