The short answer is "you are probably good to go"
The more elaborate answer is "it depends"
The depends part is based on the likely loading of the table top. An ideal tool for helping with the decision is Sagulator as @Graphus mentioned in his comment.
Before you use Sagulator, realize that it is based on structural engineering principles and guidelines. The allowable sag is determined by aesthetic considerations and even the "borderlne" sag is arrived at by the deflection being in excess of 1/360 of the span. Borderline does not usually mean that the structure is in imminent danger of failing catostrophically (i.e. breaking). When you get to "excessive" sag there is usually danger that the structure will actually fracture.
In computer programming there is an old axiom - GIGO - Garbage In Garbage Out. The same is true of using Sagulator which requires that only if you make all of the data selections accurately and knowledgably will you get a reliable answer.
The first piece of information that is needed is the Shelf Material. This allows Sagulator to determine the material stiffness of the wood by selecting a value for its modulus of elasticity. Two pieces of wood with identical dimensions will vary markedly in their stiffness (resistance to bending) depending on their species.
The next item, shelf Attachment is a tricky one and should not be overlooked. Engineers refers to this as "end conditions" which are either fixed or pinned ends. In steel structures a fixed end is welded and even a connection with many rivets is considered a pinned end. It is very difficult to get an end condition in wood that would be considered fixed (perhaps a well executed dove-tail might fit the bill.) The safest and sanest choice is the "floating" option because it is actually referring to "pinned". The "attached" option actually refers to a fixed end which in wood approaches being mythical. The difference in calculated results will be be in the range of 400% difference in calculated sag (e.g. 0.02 instead of 0.10.)
Shelf load is pretty straight forward and it would be best to use a uniform load because then weight of the wood itself can be included easily. Not to include the weight of the shelf is to imply that a long piece of wood supported at its ends with no additional load will not sag.The dimensions of span, depth and thickness should be entered accurately.
Some times people get lazy when entering the Thickness and will enter 1" when the thickness is actually 3/4". This small difference will yield a difference of more than 125% in the expected sag of the shelf. Even using 1" instead of 7/8" would give an error of about 50%.
For the table in question I entered type of wood as sugar maple (which has the same modulus of elasticity as the sapele), floating ends, 72" span, 20" depth and 1" thickness. For loading, I allowed 5.5 pounds per foot for the table top plus another 10 pounds per foot for stuff carried by the top.
The result was a total sag of 0.15" which is quite acceptable.
But wait a minute how about the apron that is actually carrying the load (sort of)
The makers of Sagulator conveniently have an "edge strip" feature which allows for stiffening a shelf. That's exactly what the apron is doing for the table top. I entered sugar maple, 3.5" width, 1" thickness and got a total sag of 0.05" which is even more acceptable.
Having vented on Sagulator, I would like to suggest that there might be a bit of problem with geometric stability in that the end legs are only twenty inches apart (set in 10" from front and rear). I suggest that you move them closer to the edges, say 5" or 6". The cantilever at the ends of 1' should be fine. Make scale drawings to see whether it looks reasonable to you.
The legs themselves should do the job if securely fastened to apron. I assume you have something aesthetically pleasing in mind for that.
