Did LIGO account for curvature as expected from the pear shaped earth or the oblate spheroid sphere ? That is what I am trying to find out . Which shape did Bob the builder account for ? Apparently neither since he used a ellipsoid model constructed on a mythical centre of the earth.

An oblate spheroid is a surface of revolution obtained by rotating an ellipse about its minor axis, so it is a form of ellipsoid and indeed it is sometimes called an oblate ellipsoid. They mean the same thing in this case.

We know that WSG-84 is nothing but a mathematical construct in which earths centre of gravity is the starting reference point for the coordinate system .

Yes of course. It's the closest approximation we have for the true shape of the Earth which is also a simply defined mathematical shape. Its accuracy varies depending on where you are, but is perfectly acceptable for GPS etc. For many purposes, a simple sphere would do, but this is just a better approximation.

I mean read this quote from the conclusion on this link https://gisgeography.com/wgs84-world-geodetic-system/

It states "Never before have we’ve been able to estimate the ellipsoid with such precision." Ho ho - estimate with precision .

Every physical real world measurement is an approximation within acceptable tolerances, so yeah, estimate with precision. Parts for racing engines are made with high precision, but they're never exact, they are either within tolerance or not.

Where is that centre of gravity on the pear shape /oblate model . Read the links I posted and you will see that is a big problem . There is no centre of gravity .

LIGO constructed its arms plane and level according to normal plane surveying techniques . That's my view .

Well except that the people who built it are telling you they used WGS-84. You have the evidence, you quote from the source. Do you accept it?