That doesn't really answer the question though. Would a private railway be required to cost the same?
The standards would have to be the same, and to a very great extent the standards set the cost.
You could ask ‘would a private railway have to meet the same standards’ - to which the answer is ‘it depends what sort of railway you want’
How much infrastructure costs depends very much on the specification that is set.
If it was easy to lay and maintain track, don’t you think all the heritage railways would have extended their lines a lot more by now?
So the simple answer, is yes, a private company could do it cheaper. If there was no connection to the Network Rail network, they don’t even have to stick to standard gauge. Similarly, if there was only ever one ‘engine in steam’, you don’t need a very complex signalling system.
But of course, if compromises are made, through running of trains becomes harder.
So you really have to compare on a like for like cost. A conventional scheme (becomes it’s not always Network Rail that actually does the work, or runs the scheme) will have advantages and disadvantages. A private company will also have advantages and disadvantages. It’s not always possible to work out in advance which, in the end, will be the lower cost option.