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We study feebly compact shift-continuous $T_1$-topologies on the symmetricinverse semigroup $\mathscr{I}_\lambda^n$ of finite transformations of the rank$\leqslant n$. For any positive integer $n\geqslant2$ and any infinite cardinal$\lambda$ a Hausdorff countably pracompact non-compact shift-continuoustopology on $\mathscr{I}_\lambda^n$ is constructed. We show that for anarbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ fora $T_1$-topology $\tau$ on $\mathscr{I}_\lambda^n$ the following conditions areequivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feeblycompact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$$\left(\mathscr{I}_\lambda^n,\tau\right)$ is H-closed; $(v)$$\left(\mathscr{I}_\lambda^n,\tau\right)$ is$\mathbb{N}_{\mathfrak{d}}$-compact for the discrete countable space$\mathbb{N}_{\mathfrak{d}}$; $(vi)$ $\left(\mathscr{I}_\lambda^n,\tau\right)$is $\mathbb{R}$-compact; $(vii)$ $\left(\mathscr{I}_\lambda^n,\tau\right)$ isinfra H-closed. Also we prove that for an arbitrary positive integer $n$ and anarbitrary infinite cardinal $\lambda$ every shift-continuous semiregular feeblycompact $T_1$-topology $\tau$ on $\mathscr{I}_\lambda^n$ is compact.

On feebly compact semitopological symmetric inverse semigroups of a bounded finite rank