Can "weakly shrewd" cardinals exist below the continuum?

Lévy introduced weak $\Pi^n_m$-indescribability as a natural weakening of, well, $\Pi^n_m$-indescribability (he introduced the two at the same time). More recently, Rathjen has introduced shrewd cardinals as an extension of $\Pi^n_m$-indescribability to transfinite $n$. Say that a cardinal $\kappa$ is weakly $\lambda$-shrewd iff for any unary predicate $\phi$ in the language of set theory plus an unary predicate symbol, and any set $A\subseteq\kappa$ with $(\mathcal{P}^\lambda(\kappa);\in, A)\models\phi$ there is $\lambda'$ and $\kappa'<\kappa$ such that $(\mathcal{P}^{\lambda'}(\kappa');\in, A\cap \kappa')\models\phi$. Here $\mathcal{P}^\lambda$ is iterated powerset with unions at limits. A fairly standard coding argument shows that a strong limit weakly $\lambda$-shrewd cardinal is just $\lambda$-shrewd. So we have the typical presentation for smaller large cardinals wherein the weak version plus being a strong limit equals the strong version (e.g. with inaccessible and Mahlo). There is an argument that weak $1$-shrewdness is compatible with being less than the continuum; though I don't know what this argument is (maybe just Cohen forcing?). It seems plausible to me that the argument breaks when attempting to generalize to higher $\lambda$, though I hope it doesn't. Is full weak shrewdness compatible with being less than the continuum?