Thermal convection needs only heat and gravity to work and it doesn't work very well horizontally.
On the other hand, convection results in a gradient, with the cool, down-flowing medium getting warmer towards the bottom.
A horizontal core with a vertical convention going across it would probably be a preferable solution, then: each and every segment of that core would be enjoying the same sort of convection cooling.
Perhaps gravity would be useful in emergency shutdowns? With the matter tank on top and the antimatter tank at the very bottom, an interruption of fuel flow would result in the vertical core being drowned in inert matter raining from above, while the reactive antimatter drained to the bottom. Whether this would do any good in practice, it's hard to tell. But it bears some resemblance to passive safety measures taken in nuclear power plants today.