A mobile agent, starting from a node $s$ of a simple undirected connected graph $G=(V,E)$, has to explore all nodes and edges of $G$ using the minimum number of edge traversals. To do so, the agent uses a deterministic algorithm that allows it to gain information on $G$ as it traverses its edges. During its exploration, the agent must always respect the constraint of knowing a path of length at most $D$ to go back to node $s$. The upper bound $D$ is fixed as being equal to $(1+\alpha)r$, where $r$ is the eccentricity of node $s$ (i.e., the maximum distance from $s$ to any other node) and $\alpha$ is any positive real constant. This task has been introduced by Duncan et al. [ACM Trans. Algorithms 2006] and is known as \emph{distance-constrained exploration}. The \emph{penalty} of an exploration algorithm running in $G$ is the number of edge traversals made by the agent in excess of $|E|$. Panaite and Pelc [J. Algorithms 1999] gave an algorithm for solving exploration without any constraint on the moves that is guaranteed to work in every graph $G$ with a (small) penalty in $\mathcal{O}(|V|)$. Hence, a natural question is whether we could obtain a distance-constrained exploration algorithm with the same guarantee as well. In this paper, we provide a negative answer to this question. We also observe that an algorithm working in every graph $G$ with a linear penalty in $|V|$ cannot be obtained for the task of \emph{fuel-constrained exploration}, another variant studied in the literature. This solves an open problem posed by Duncan et al. [ACM Trans. Algorithms 2006] and shows a fundamental separation with the task of exploration without constraint on the moves.