Weak gravitational lensing mass mapping offers a direct probe of the matter distribution. Accurate reconstruction of mass maps from masked shear catalogs remains challenging due to survey boundaries and spatially varying noise. In AKRA 2.0, we addressed the mask problem on the curved sky by constructing and inverting the normal-equation matrix $\bf{H} \equiv \mathbf{A}^\mathrm{T}\mathbf{N}^{-1} \mathbf{A}$ explicitly, necessitating a split-scale strategy that reconstructed different angular scales independently to reach high resolution. Here we present AKRA 3.0, in which $𝐇$ is treated as a linear operator and the normal equations are solved by the conjugate gradient (CG) method. This reformulation reduces the memory requirement from $𝒪(N^2)$ to $𝒪(N)$ and the inversion cost from $𝒪(N^3)$ to $\mathcal{O}(N_{\rm iter}N^{3/2}), N \sim \ell_{\rm{max}}^2$ for full-sky (SHT-based) operations. Such optimizations render high-resolution full-sky reconstruction tractable for Stage~III and Stage~IV surveys. Applying AKRA 3.0 to the DES Y3 catalog, we produce the highest-resolution convergence map of this dataset to date at HEALPix $N_{\rm{nside}}= 2048$ without imposing a cosmological signal prior. We validate the map reconstruction and its auto- and cross-power responses using end-to-end simulations with known input fields and present a preliminary power-spectrum analysis of the DES~Y3 maps. The reconstructed E-mode convergence map will be made publicly available to facilitate future cosmological studies.