After reviewing the protocol-dependent properties of HDA, which thermally anneals to LDA, and the data gap over an unusually large T-range between HDA, LDA, and water, we investigate whether or not, despite HDA's ill-defined state and distinction from a glass, the HDL-LDL fluctuations view of the two-liquid model can explain water's anomalous behavior. An analysis of the density, rho, compressibility, beta, heat capacity, C-p, and thermal conductivity, kappa, of water over a monotonic (continuous) path bridging this data gap shows the following: (i) Such a path between rho(water) at 320 K and rho(HDA) yields an untenable thermal expansion coefficient of water. (ii) There is neither a continuous path between beta(water) at 353 K and beta(HDA), nor between C-p,C-water at 363 K and C-p,C-HDA. (iii) The same value of rho(water), of beta(water), or of C-p,C-water at two temperatures separated by a maxima or a minima is incompatible with the HDL-LDL fluctuations view. (iv) kappa(LDA) at similar to 140 K is about twice that of kappa(water) at 253 K. (v) kappa(HDA) at 120 K is incompatible with kappa(water) at T > 320 K. Thus, there is an internal inconsistency between the thermodynamics of HDA seen as a glass and that of water seen as an HDL-LDL mixture, which is incompatible with both the HDL-LDL fluctuations view and the liquid-liquid transition.