We address Multi-Robot Exploration and Relaying (MRER): a team of robots must explore an unknown environment and deliver acquired information to a fixed base station within a mission time limit. The central challenge is deciding when each robot should stop exploring and relay: this depends on what the robot is likely to find ahead, what information it uniquely holds, and whether immediate or future delivery is more valuable. Prior approaches either ignore the reporting requirement entirely or rely on fixed-schedule relay strategies that cannot adapt to environment structure, team composition, or mission progress. We introduce PRoID (Predicted Rate of Information Delivery), a relay criterion that uses learned map prediction to estimate each robot's future information gain along its planned path, accounting for what teammates are already relaying. PRoID triggers relay when immediate return yields higher information delivery per unit time. We further propose PRoID-Safe, a failure-aware extension that incorporates robot survival probability into the relay criterion, naturally biasing decisions toward earlier relay as failure risk grows. We evaluate on real-world indoor floor plan datasets and show that PRoID and PRoID-Safe outperform fixed-schedule baselines, with stronger relative gains in failure scenarios.