Step 1: The clinical aim is to find contrast escaping through a hole in the esophageal wall. The safety of the agent if it leaks decides which contrast is used.
Step 2: Barium is excellent for mucosal detail but is disastrous in the mediastinum, where extravasated barium triggers a chemical mediastinitis. Therefore, when a perforation is on the table, the first-line agent must be water-soluble.
Step 3: Iohexol is exactly that - a non-ionic, low-osmolar, water-soluble iodinated contrast (Omnipaque) that is absorbed safely if it spills outside the lumen. So it is the dye chosen to evaluate a suspected perforation.
Step 4: The others fail the test: barium sulphate is hazardous on leak, gadolinium is an MRI agent not used for a fluoroscopic swallow here, and the generic "iodine dye" is too vague next to the specifically appropriate iohexol.
\[\boxed{\text{Iohexol}}\]